We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^3)).

Strict Trs:
  { terms(N) -> cons(recip(sqr(N)))
  , sqr(0()) -> 0()
  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
  , add(0(), X) -> X
  , add(s(X), Y) -> s(add(X, Y))
  , dbl(0()) -> 0()
  , dbl(s(X)) -> s(s(dbl(X)))
  , first(0(), X) -> nil()
  , first(s(X), cons(Y)) -> cons(Y) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^3))

We add the following dependency tuples:

Strict DPs:
  { terms^#(N) -> c_1(sqr^#(N))
  , sqr^#(0()) -> c_2()
  , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X))
  , add^#(0(), X) -> c_4()
  , add^#(s(X), Y) -> c_5(add^#(X, Y))
  , dbl^#(0()) -> c_6()
  , dbl^#(s(X)) -> c_7(dbl^#(X))
  , first^#(0(), X) -> c_8()
  , first^#(s(X), cons(Y)) -> c_9() }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^3)).

Strict DPs:
  { terms^#(N) -> c_1(sqr^#(N))
  , sqr^#(0()) -> c_2()
  , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X))
  , add^#(0(), X) -> c_4()
  , add^#(s(X), Y) -> c_5(add^#(X, Y))
  , dbl^#(0()) -> c_6()
  , dbl^#(s(X)) -> c_7(dbl^#(X))
  , first^#(0(), X) -> c_8()
  , first^#(s(X), cons(Y)) -> c_9() }
Weak Trs:
  { terms(N) -> cons(recip(sqr(N)))
  , sqr(0()) -> 0()
  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
  , add(0(), X) -> X
  , add(s(X), Y) -> s(add(X, Y))
  , dbl(0()) -> 0()
  , dbl(s(X)) -> s(s(dbl(X)))
  , first(0(), X) -> nil()
  , first(s(X), cons(Y)) -> cons(Y) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^3))

We estimate the number of application of {2,4,6,8,9} by
applications of Pre({2,4,6,8,9}) = {1,3,5,7}. Here rules are
labeled as follows:

  DPs:
    { 1: terms^#(N) -> c_1(sqr^#(N))
    , 2: sqr^#(0()) -> c_2()
    , 3: sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X))
    , 4: add^#(0(), X) -> c_4()
    , 5: add^#(s(X), Y) -> c_5(add^#(X, Y))
    , 6: dbl^#(0()) -> c_6()
    , 7: dbl^#(s(X)) -> c_7(dbl^#(X))
    , 8: first^#(0(), X) -> c_8()
    , 9: first^#(s(X), cons(Y)) -> c_9() }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^3)).

Strict DPs:
  { terms^#(N) -> c_1(sqr^#(N))
  , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X))
  , add^#(s(X), Y) -> c_5(add^#(X, Y))
  , dbl^#(s(X)) -> c_7(dbl^#(X)) }
Weak DPs:
  { sqr^#(0()) -> c_2()
  , add^#(0(), X) -> c_4()
  , dbl^#(0()) -> c_6()
  , first^#(0(), X) -> c_8()
  , first^#(s(X), cons(Y)) -> c_9() }
Weak Trs:
  { terms(N) -> cons(recip(sqr(N)))
  , sqr(0()) -> 0()
  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
  , add(0(), X) -> X
  , add(s(X), Y) -> s(add(X, Y))
  , dbl(0()) -> 0()
  , dbl(s(X)) -> s(s(dbl(X)))
  , first(0(), X) -> nil()
  , first(s(X), cons(Y)) -> cons(Y) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^3))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ sqr^#(0()) -> c_2()
, add^#(0(), X) -> c_4()
, dbl^#(0()) -> c_6()
, first^#(0(), X) -> c_8()
, first^#(s(X), cons(Y)) -> c_9() }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^3)).

Strict DPs:
  { terms^#(N) -> c_1(sqr^#(N))
  , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X))
  , add^#(s(X), Y) -> c_5(add^#(X, Y))
  , dbl^#(s(X)) -> c_7(dbl^#(X)) }
Weak Trs:
  { terms(N) -> cons(recip(sqr(N)))
  , sqr(0()) -> 0()
  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
  , add(0(), X) -> X
  , add(s(X), Y) -> s(add(X, Y))
  , dbl(0()) -> 0()
  , dbl(s(X)) -> s(s(dbl(X)))
  , first(0(), X) -> nil()
  , first(s(X), cons(Y)) -> cons(Y) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^3))

Consider the dependency graph

  1: terms^#(N) -> c_1(sqr^#(N))
     -->_1 sqr^#(s(X)) ->
           c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) :2
  
  2: sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X))
     -->_3 dbl^#(s(X)) -> c_7(dbl^#(X)) :4
     -->_1 add^#(s(X), Y) -> c_5(add^#(X, Y)) :3
     -->_2 sqr^#(s(X)) ->
           c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) :2
  
  3: add^#(s(X), Y) -> c_5(add^#(X, Y))
     -->_1 add^#(s(X), Y) -> c_5(add^#(X, Y)) :3
  
  4: dbl^#(s(X)) -> c_7(dbl^#(X))
     -->_1 dbl^#(s(X)) -> c_7(dbl^#(X)) :4
  

Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).

  { terms^#(N) -> c_1(sqr^#(N)) }


We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^3)).

Strict DPs:
  { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X))
  , add^#(s(X), Y) -> c_5(add^#(X, Y))
  , dbl^#(s(X)) -> c_7(dbl^#(X)) }
Weak Trs:
  { terms(N) -> cons(recip(sqr(N)))
  , sqr(0()) -> 0()
  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
  , add(0(), X) -> X
  , add(s(X), Y) -> s(add(X, Y))
  , dbl(0()) -> 0()
  , dbl(s(X)) -> s(s(dbl(X)))
  , first(0(), X) -> nil()
  , first(s(X), cons(Y)) -> cons(Y) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^3))

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { sqr(0()) -> 0()
    , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
    , add(0(), X) -> X
    , add(s(X), Y) -> s(add(X, Y))
    , dbl(0()) -> 0()
    , dbl(s(X)) -> s(s(dbl(X))) }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^3)).

Strict DPs:
  { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X))
  , add^#(s(X), Y) -> c_5(add^#(X, Y))
  , dbl^#(s(X)) -> c_7(dbl^#(X)) }
Weak Trs:
  { sqr(0()) -> 0()
  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
  , add(0(), X) -> X
  , add(s(X), Y) -> s(add(X, Y))
  , dbl(0()) -> 0()
  , dbl(s(X)) -> s(s(dbl(X))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^3))

We decompose the input problem according to the dependency graph
into the upper component

  { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) }

and lower component

  { add^#(s(X), Y) -> c_5(add^#(X, Y))
  , dbl^#(s(X)) -> c_7(dbl^#(X)) }

Further, following extension rules are added to the lower
component.

{ sqr^#(s(X)) -> sqr^#(X)
, sqr^#(s(X)) -> add^#(sqr(X), dbl(X))
, sqr^#(s(X)) -> dbl^#(X) }

TcT solves the upper component with certificate YES(O(1),O(n^1)).

Sub-proof:
----------
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(n^1)).
  
  Strict DPs:
    { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) }
  Weak Trs:
    { sqr(0()) -> 0()
    , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
    , add(0(), X) -> X
    , add(s(X), Y) -> s(add(X, Y))
    , dbl(0()) -> 0()
    , dbl(s(X)) -> s(s(dbl(X))) }
  Obligation:
    innermost runtime complexity
  Answer:
    YES(O(1),O(n^1))
  
  We use the processor 'Small Polynomial Path Order (PS,1-bounded)'
  to orient following rules strictly.
  
  DPs:
    { 1: sqr^#(s(X)) ->
         c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) }
  
  Sub-proof:
  ----------
    The input was oriented with the instance of 'Small Polynomial Path
    Order (PS,1-bounded)' as induced by the safe mapping
    
     safe(sqr) = {}, safe(0) = {}, safe(s) = {1}, safe(add) = {1, 2},
     safe(dbl) = {}, safe(sqr^#) = {}, safe(c_3) = {}, safe(add^#) = {},
     safe(dbl^#) = {}
    
    and precedence
    
     empty .
    
    Following symbols are considered recursive:
    
     {sqr^#}
    
    The recursion depth is 1.
    
    Further, following argument filtering is employed:
    
     pi(sqr) = [], pi(0) = [], pi(s) = [1], pi(add) = [1, 2],
     pi(dbl) = 1, pi(sqr^#) = [1], pi(c_3) = [1, 2, 3], pi(add^#) = [],
     pi(dbl^#) = []
    
    Usable defined function symbols are a subset of:
    
     {sqr^#, add^#, dbl^#}
    
    For your convenience, here are the satisfied ordering constraints:
    
      pi(sqr^#(s(X))) = sqr^#(s(; X);)                                    
                      > c_3(add^#(),  sqr^#(X;),  dbl^#();)               
                      = pi(c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)))
                                                                          
  
  The strictly oriented rules are moved into the weak component.
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(1)).
  
  Weak DPs:
    { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) }
  Weak Trs:
    { sqr(0()) -> 0()
    , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
    , add(0(), X) -> X
    , add(s(X), Y) -> s(add(X, Y))
    , dbl(0()) -> 0()
    , dbl(s(X)) -> s(s(dbl(X))) }
  Obligation:
    innermost runtime complexity
  Answer:
    YES(O(1),O(1))
  
  The following weak DPs constitute a sub-graph of the DG that is
  closed under successors. The DPs are removed.
  
  { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) }
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(1)).
  
  Weak Trs:
    { sqr(0()) -> 0()
    , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
    , add(0(), X) -> X
    , add(s(X), Y) -> s(add(X, Y))
    , dbl(0()) -> 0()
    , dbl(s(X)) -> s(s(dbl(X))) }
  Obligation:
    innermost runtime complexity
  Answer:
    YES(O(1),O(1))
  
  No rule is usable, rules are removed from the input problem.
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(1)).
  
  Rules: Empty
  Obligation:
    innermost runtime complexity
  Answer:
    YES(O(1),O(1))
  
  Empty rules are trivially bounded

We return to the main proof.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict DPs:
  { add^#(s(X), Y) -> c_5(add^#(X, Y))
  , dbl^#(s(X)) -> c_7(dbl^#(X)) }
Weak DPs:
  { sqr^#(s(X)) -> sqr^#(X)
  , sqr^#(s(X)) -> add^#(sqr(X), dbl(X))
  , sqr^#(s(X)) -> dbl^#(X) }
Weak Trs:
  { sqr(0()) -> 0()
  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
  , add(0(), X) -> X
  , add(s(X), Y) -> s(add(X, Y))
  , dbl(0()) -> 0()
  , dbl(s(X)) -> s(s(dbl(X))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.

DPs:
  { 2: dbl^#(s(X)) -> c_7(dbl^#(X))
  , 3: sqr^#(s(X)) -> sqr^#(X)
  , 4: sqr^#(s(X)) -> add^#(sqr(X), dbl(X))
  , 5: sqr^#(s(X)) -> dbl^#(X) }
Trs: { dbl(0()) -> 0() }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_5) = {1}, Uargs(c_7) = {1}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
          [sqr](x1) = [0]         
                                  
                [0] = [0]         
                                  
            [s](x1) = [1] x1 + [4]
                                  
      [add](x1, x2) = [4] x2 + [0]
                                  
          [dbl](x1) = [1]         
                                  
        [sqr^#](x1) = [2] x1 + [0]
                                  
    [add^#](x1, x2) = [1]         
                                  
        [dbl^#](x1) = [1] x1 + [6]
                                  
          [c_5](x1) = [1] x1 + [0]
                                  
          [c_7](x1) = [1] x1 + [0]
  
  The order satisfies the following ordering constraints:
  
          [sqr(0())] =  [0]                     
                     >= [0]                     
                     =  [0()]                   
                                                
         [sqr(s(X))] =  [0]                     
                     ?  [8]                     
                     =  [s(add(sqr(X), dbl(X)))]
                                                
       [add(0(), X)] =  [4] X + [0]             
                     >= [1] X + [0]             
                     =  [X]                     
                                                
      [add(s(X), Y)] =  [4] Y + [0]             
                     ?  [4] Y + [4]             
                     =  [s(add(X, Y))]          
                                                
          [dbl(0())] =  [1]                     
                     >  [0]                     
                     =  [0()]                   
                                                
         [dbl(s(X))] =  [1]                     
                     ?  [9]                     
                     =  [s(s(dbl(X)))]          
                                                
       [sqr^#(s(X))] =  [2] X + [8]             
                     >  [2] X + [0]             
                     =  [sqr^#(X)]              
                                                
       [sqr^#(s(X))] =  [2] X + [8]             
                     >  [1]                     
                     =  [add^#(sqr(X), dbl(X))] 
                                                
       [sqr^#(s(X))] =  [2] X + [8]             
                     >  [1] X + [6]             
                     =  [dbl^#(X)]              
                                                
    [add^#(s(X), Y)] =  [1]                     
                     >= [1]                     
                     =  [c_5(add^#(X, Y))]      
                                                
       [dbl^#(s(X))] =  [1] X + [10]            
                     >  [1] X + [6]             
                     =  [c_7(dbl^#(X))]         
                                                

The strictly oriented rules are moved into the weak component.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict DPs: { add^#(s(X), Y) -> c_5(add^#(X, Y)) }
Weak DPs:
  { sqr^#(s(X)) -> sqr^#(X)
  , sqr^#(s(X)) -> add^#(sqr(X), dbl(X))
  , sqr^#(s(X)) -> dbl^#(X)
  , dbl^#(s(X)) -> c_7(dbl^#(X)) }
Weak Trs:
  { sqr(0()) -> 0()
  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
  , add(0(), X) -> X
  , add(s(X), Y) -> s(add(X, Y))
  , dbl(0()) -> 0()
  , dbl(s(X)) -> s(s(dbl(X))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ sqr^#(s(X)) -> dbl^#(X)
, dbl^#(s(X)) -> c_7(dbl^#(X)) }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict DPs: { add^#(s(X), Y) -> c_5(add^#(X, Y)) }
Weak DPs:
  { sqr^#(s(X)) -> sqr^#(X)
  , sqr^#(s(X)) -> add^#(sqr(X), dbl(X)) }
Weak Trs:
  { sqr(0()) -> 0()
  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
  , add(0(), X) -> X
  , add(s(X), Y) -> s(add(X, Y))
  , dbl(0()) -> 0()
  , dbl(s(X)) -> s(s(dbl(X))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

We use the processor 'custom shape polynomial interpretation' to
orient following rules strictly.

DPs:
  { 1: add^#(s(X), Y) -> c_5(add^#(X, Y))
  , 2: sqr^#(s(X)) -> sqr^#(X)
  , 3: sqr^#(s(X)) -> add^#(sqr(X), dbl(X)) }
Trs:
  { sqr(0()) -> 0()
  , dbl(0()) -> 0() }

Sub-proof:
----------
  The following argument positions are considered usable:
    Uargs(c_5) = {1}
  TcT has computed the following constructor-restricted polynomial
  interpretation.
        [sqr](x1) = 1 + x1^2
                            
            [0]() = 0       
                            
          [s](x1) = 2 + x1  
                            
    [add](x1, x2) = x1 + x2 
                            
        [dbl](x1) = 2 + 2*x1
                            
      [sqr^#](x1) = 3 + x1^2
                            
  [add^#](x1, x2) = 2 + x1  
                            
      [dbl^#](x1) = 0       
                            
        [c_5](x1) = x1      
                            
        [c_7](x1) = 0       
                            
  
  This order satisfies the following ordering constraints.
  
          [sqr(0())] =  1                       
                     >                          
                     =  [0()]                   
                                                
         [sqr(s(X))] =  5 + 4*X + X^2           
                     >= 5 + X^2 + 2*X           
                     =  [s(add(sqr(X), dbl(X)))]
                                                
       [add(0(), X)] =  X                       
                     >= X                       
                     =  [X]                     
                                                
      [add(s(X), Y)] =  2 + X + Y               
                     >= 2 + X + Y               
                     =  [s(add(X, Y))]          
                                                
          [dbl(0())] =  2                       
                     >                          
                     =  [0()]                   
                                                
         [dbl(s(X))] =  6 + 2*X                 
                     >= 6 + 2*X                 
                     =  [s(s(dbl(X)))]          
                                                
       [sqr^#(s(X))] =  7 + 4*X + X^2           
                     >  3 + X^2                 
                     =  [sqr^#(X)]              
                                                
       [sqr^#(s(X))] =  7 + 4*X + X^2           
                     >  3 + X^2                 
                     =  [add^#(sqr(X), dbl(X))] 
                                                
    [add^#(s(X), Y)] =  4 + X                   
                     >  2 + X                   
                     =  [c_5(add^#(X, Y))]      
                                                

The strictly oriented rules are moved into the weak component.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Weak DPs:
  { sqr^#(s(X)) -> sqr^#(X)
  , sqr^#(s(X)) -> add^#(sqr(X), dbl(X))
  , add^#(s(X), Y) -> c_5(add^#(X, Y)) }
Weak Trs:
  { sqr(0()) -> 0()
  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
  , add(0(), X) -> X
  , add(s(X), Y) -> s(add(X, Y))
  , dbl(0()) -> 0()
  , dbl(s(X)) -> s(s(dbl(X))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ sqr^#(s(X)) -> sqr^#(X)
, sqr^#(s(X)) -> add^#(sqr(X), dbl(X))
, add^#(s(X), Y) -> c_5(add^#(X, Y)) }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Weak Trs:
  { sqr(0()) -> 0()
  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
  , add(0(), X) -> X
  , add(s(X), Y) -> s(add(X, Y))
  , dbl(0()) -> 0()
  , dbl(s(X)) -> s(s(dbl(X))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

No rule is usable, rules are removed from the input problem.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Rules: Empty
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

Hurray, we answered YES(O(1),O(n^3))